Number Theory(congruance)

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Hasib
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Number Theory(congruance)

Unread post by Hasib » Fri Dec 10, 2010 2:53 pm

If $x^3+y^3=z^3$ Show that one of the three must be a multiple of 7
Last edited by Hasib on Fri Dec 10, 2010 10:10 pm, edited 5 times in total.
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Moon
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Re: Number Theory(congruance)

Unread post by Moon » Fri Dec 10, 2010 3:49 pm

Hasib, why don't you consider using LaTeX? Now, you can write equations without knowing it!
viewtopic.php?f=9&t=54&p=175#p175
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Please install LaTeX fonts in your PC for better looking equations,
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Hasib
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Re: Number Theory(congruance)

Unread post by Hasib » Fri Dec 10, 2010 10:11 pm

Why my code doesnt working?
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Moon
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Re: Number Theory(congruance)

Unread post by Moon » Fri Dec 10, 2010 10:26 pm

Hasib, I can see your equation just fine. Please clean the cache of your browser and try again. If you are using firefox then go to Tools>Options>Advanced then click "Clear Now".

Let me know if that solves your problem. Thanks.
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Please install LaTeX fonts in your PC for better looking equations,
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Masum
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Re: Number Theory(congruance)

Unread post by Masum » Fri Dec 10, 2010 10:44 pm

We use the fact $a^3\equiv 0,\pm 1\ (mod\ 7)$
If none of $x,y,z$ divisible by $7,$then we have:
$x^3+y^3\equiv \pm(1+1)\equiv \pm 2\ (mod\ 7)$ but $z^3$ is not $\equiv \pm 2\ (mod\ 7)$.Contradiction
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Hasib
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Re: Number Theory(congruance)

Unread post by Hasib » Sat Dec 11, 2010 6:16 pm

ThanQ Moon vaia /clubsuit
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nmmahi
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Re: Number Theory(congruance)

Unread post by nmmahi » Sun Dec 12, 2010 8:08 pm

Dear Hasib,are you considering $x^3+y^3=z^3$?

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Re: Number Theory(congruance)

Unread post by Hasib » Sun Dec 12, 2010 10:49 pm

@nmmahi: yeah :)\[\clubsuit^{\clubsuit\clubsuit}\clubsuit\]
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Re: Number Theory(congruance)

Unread post by Hasib » Mon Dec 13, 2010 5:52 pm

@Masum vai: i think, this problem is very interesting. Cause, with the Fermat's last theorem we know there is no positive solution of this form \[a^n+b^n=c^n\] whence \[n\ge 3\]
so, how we prove this one? I think it's a great DOSS! I take the problem from The Art and Craft of Problem Solving.\[\clubsuit^\clubsuit\clubsuit\]
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Re: Number Theory(congruance)

Unread post by kamrul2010 » Thu Dec 16, 2010 1:02 am

খাইসে...মজার জিনিস তো! :D
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